The puzzle asks for 0<k<z, z=>16. The given solution n=833021343
is the
smallest for z=16 (maybe the intended question was 0<k<=z ?).
The first three with z>16 all end at z=17 and are: 12465115083 (11
digits),
2735670562857 (13 digits), 83003201588773 (14 digits). The first of
these is
surprisingly small; around 1/6659 of the third.
The smallest n for z = 2 to 19 is:
1, 1, 3, 7, 7, 15, 37, 163, 177, 4503, 4503, 4503, 4503, 4503,
833021343,
12465115083, 95854279610863, 1158174141556287.
I will submit the extension to
http://www.research.att.com/~njas/sequences/A130003.

***

Bernardo wrote:

12465115083 yields k=16. I have not computed the chance for a
given
number of n digit to be in the sequence, but I suspect it drops to
zero
quickly enough to expect only a finite number of solutions. I was
actually already surprised to find the solution for k=16 so
relatively
close to the one by Farideh. There are no better solutions below
284*10^9.

***

Fred wrote:

12465115083 is the smallest natural number n such that n+4^k is
prime
for 0<k<=z, z=>16.
I ran an exhaustive search, iterating through primes p = n+4^16 and
then attempting to fill in the gaps.